Apparatus for teaching mathematics



3 Sheets-Sheet l M. B. SLATER APPARATUS FOR TEACHING MATHEMATICS May 5,1964 Filed Jan. 5, 1961 7 2 x 2 x A a 3 ZJIlIII,

My m 0.0 kw M FF Q Z l l f e m f m m W mu my HM r wm w 6 WM w w G F a wL APPARATUS FOR TEACHING MATHEMATICS Filed Jan. 5, 1961 3 Sheets-Sheet 3Nam United States Patent 3,131,488 APPARATUS FOR TEACHlNG MATHEMATICSMarguerite ll. Slater, 139 E. 36th St, New York 16, N31. Filed Jan. 5,1961, Ser. No. 80,949 8 Claims. (Cl. 35-31) This invention relates toeducational apparatus for learning mathematics, and more particularlythe elementary facts of addition and subtraction in arithmetic.

The first few steps in addition and subtraction are generally firstlearned by counting, that is by tallying on ones fingers or using stonesor other objects. This is similar to using an abacus. Often children arenext taught to memorize most of the addition or subtraction facts solelyby roteby repetition of tables-with little, if any concept andconnotations of the meaning of the words or figures they repeat orallyor in writing.

In order to make progress in mathematics, it is necessary to advanceboth from rote and from the use of concrete aids in addition orsubtraction to an understanding use of abstract symbols without tallyingor counting. There has been a lack of apparatus for helping the teacherto convey to the child the relation between objects and abstractlystated addition and subtraction facts.

Unfortunately, many intelligent people never advance in the knowledge ofaddition beyond the stage of counting on ones fingers. Not only childrenin the higher grades, but a very large proportion of college studentshave given evidence, by unmistakable signs, of still using counting whenanything arithmetical comes up. This lack of full understanding of theelementary facts of addi tion and subtraction is a great handicap whenmore complex mathematical operations must be dealt with in everydaylife, and an ever greater handicap in work in the sciences.

The new educational apparatus which forms the subject of this inventionfacilitates the transition from adding by counting to a thoroughknowledge of all the ele mentary addition and subtraction facts, and touse of abstract symbols therefor with understanding. It is useful inother mathematical connections.

When I speak of simple or elementary addition and subtraction facts,reference is to the classification, common in education, where not morethan two quantities are added together or subtracted from one another.Any instance where three or more aggregates are added together is knownas a multiple addition fact and is not an elementary addition fact.

One object of this invention is to provide teaching apparatus whichhelps the pupil make the transition from counting concrete objects to anunderstanding of abstract simple addition and subtraction facts and alsoof the facts about symbols.

Another object of the present invention is to provide apparatus to soguide the pupils mind that he will of necessity learn the elementmyaddition and subtraction facts and their relation to each other and tothe symbols therefor, so that the pupil will become so facile inhandling these elements of mathematics that he can go on to highermathematics without having to use counting or tallying.

Another object of the invention is to help the child to understand themeaning of the cardinal, as distinguished from the ordinal numbers.

Another object is to provide apparatus by the use of which a teacher cantest the pupil and know whether the pupil has progressed to a knowledgeof the relation of abstract addition and subtraction facts, asdistinguished from mere rote repetition of tables.

It is characteristic of the apparatus when used for elementary additionand subtraction facts made in accordance with my invention that there isa workspace marked off in arbitrary units totalling the sum to belearned and tiles based on the same unit such that there is at least3,l3l,483 Patented May 5, 1964 one grouping of not more than two tileswhich fill the workspace and also equal the sum to be learned.

In the drawings:

FIGURE 1 shows a plan view of a preferred form of a set of apparatus forteaching the elementary facts of addition and subtraction about thenumber 2.

FIG. 2 is a view similar to FIG. 1 of a set for teaching the facts aboutthe number 3.

FIG. 3 is a view similar to FIGS. 1 and 2 for teaching the facts aboutthe number 5.

FIG. 4 is a view similar to FIGS. 13 of apparatus for teaching the factsabout the number 8.

FIG. 5 is a plan view of an additional element for learning the factsabout the number 5 as shown in FIG. 3. It is a separate sheet or cardcarrying three extra representations of the workspace of FIG. 3 suitablefor the pupil to fill in to show the different tile groupings equallingthe number 5. They are shown filled in. This is called a semi-concreteelement.

FIG. 6 is a plan view of a modified form of board showing a workspacefor learning the number 5 and'word statements of the addition factsabout that number.

FIG. 7 is a plan view of a further set of elements for learning thefacts about the number 5, these consisting of a workspace, number symbolstatements of the facts and three extra blank representations of theworkspace. This apparatus also differs from FIGS. 1 to 4 in that theboard is composed of several parts detachably fastened together. Theworkspace, statements of facts etc. are on different puts.

FIG. 8 is a perspective view of an open loose leaf book including setsof elements for different numbers similar to those shown in FIGURES 1 to4.

FIG. 9 is a plan view of an assortment of tiles having from one toeighteen units.

FIG. 10 is a chart showing a summary of addition and subtraction factsto be taught by means of apparatus such as shown in FIGS. 19 inclusive.

FIG. 11 is a view similar to FIG. 1 of a set for teaching the factsabout the number 1.

The set of apparatus for each number comprises an element which is asurface having a workspace and an element comprising one or more tilesto fit into the workspace and fill it. Each workspace has one tile ofthe exact shape and size to fit it. This corresponds to the groupings inwhich one of the two quantities in a statement of fact is a zero. Theother tiles are lesser in area.

In this description the two quantities or numbers and their relation tothe sum or number to be learned are referred to as a statement of fact.Such a statement is divided into the sum or answer on the one hand andwhat is referred to as a grouping of the two numbers on the other. Thecharacter zero is considered as one member of the grouping in anystatement where zero is involved.

The simple sets of elements of FIGS. -1 to 4 will first be described.There is a set of elements for each sum or number to be learned.

In \FIG. 1 We have chosen to show the elements for learning the factsabout the number two. There is a surface or board 29, having thereon aworkspace 26. The boards and Workspaces in FIGS. 1-1, 2, 3, and 4 forthe numbers one, three, five and eight are numbered 200', ill, 22, 2A3and '39, 27, 28, 2? respectively. Each workspace is different in areaand represents the aggregate, number, sum or quantity to be taught.

The surface may be any convenient surface such for example as a portableboard, or table top, part of a floor or part of a wall. In the preferredform the surfaces are sheets of cardboard, plastic or light metal suchas aluminum or magnesium. If desired the board can be divided into partsdetachable from each other. Each part may contain a separate element ofthe combination. A convenient size for a sheet is 8 /2 by 11 inches. Thesheets may, if desired, be perforated for including in a loose eatbinder (see FIG. 8).

The surface may be of any size and may be or" any material such as wood,cardboard, metal or synthetic materials.

The workspace may be, for example, a depression in the board, a raisedarea or a space having its perimeter defined in any desired manner fromthe rest of the board. This workspace is divided by partition lines ormarks 36 into one or more arbitrary area units totaling the aggregatesurn about which the facts of addition or subtraction are to be learned.in FZGURE 1 the aggregate sum is two, and the workspace is divided intotwo arbitrary units. if desired an abstract symbol such as the numeralbeing taught may be placed on the final one of the arbitrary units. InFIG. 1 this is identified by the sum reference character 4 Thisworkspace is the first element in the set for a number to be taught.

The second element in each set is the removable tiles. For each setthere are one or more tiles whose area also is based on the samearbitrary unit and shaped correspondingly. In FIG. 1 three tiles 44, 45are provided. The arbitrary unit or area into which the workspace andthe tiles are divided can be considered as corresponding to the smallestwhole number. The workspace plus the tiles make up a basic set ofapparatus for teaching one aggregate stun. This :also applies even ifthe mathematical fact to be learned is not elementary addition orsubtraction. in such a case, the smallest whole number might be someother aggregate or symbol.

The tiles can be of the same material as the boa-rd, or of di ferentmaterial. The chief requirement for the tiles is that they be of a sizeand weight such that the child or other pupil who is learning can easilytake them in his hand.

There are two discrete tiles -44 in the set'for teaching the facts aboutthe quantity t-wo (FIG. 1}, and one tile 45. Each tile 54 is the samesize and shape as the basic area unit of the workspace. The tile 45comprises two such area units placed in the same relation to each otheras the two units in the workspace. Except in the tiles for the aggregatesums or quantity 2, there is no set of elements in which there are twotiles consisting of a single discrete unit.

PEG. 2 is a set of apparatus for teaching the elementary addition andsubtraction facts related to the quantity 3. The abstract symbolconsisting of the number 3 in the workspace in this figure of thedrawings is marked with the-reference character 4-1. The set of elerentsincludes one tile '44 comprising the single unit, one tile 45 comprising2 units and a tile 46 comprising 3 units and having the same size andshape as the workspace 27 on the board in this figure.

FIG. 3 comprises a set for teaching the facts about the sum 5. Thenumber 5 in the workspace 2 8 bears the reference character 42. Inaddition to t e board 22 having the workspace 2S and associatedstatements of addition and subtraction facts 3'3 and 3 respectively foraggregate sum 5, this set includes tiles d4, 45, 4 6 like those in thepreceding sets, a tile 4-"7 havin g4 units, and a tile 43 having 5 unitsarranged in the same order and shape as the units in the workspace ofthis figure.

FIG. 4 shows a set'fOr teaching the facts about the aggregate sum 8. Itincludes the board 23, the workspace 29 having eight units and a totalof 9 tiles including one each of tiles 44, 45, 46 and 48, and two oftiles 47 ike the tiles shovm in the preceding figures, and tiles 4?, 5tand 51 having 6, 7, and 8 units respectively. The number 8 in theworkspace is identified by the number 43. The addition and subtractionfacts are shown at 34 and 38 respectively.

While in the preferred example given in FlGURES l to 4 an embodiment hasbeen used which involves three elements workspace, tiles and the numbersymbol statements of the addition and subtraction facts, it should benoted that the first two elements alone have some of the values of theinvention without the number symbols.

FIGURE 8 is a perspective view of a loose leaf book 6'9 containing amultiplicity of sets each including a boa-rd and a set of tiles forteaching a given aggregate quantity or sum, one or pairs of which tileswill illustrate each of the simple addition facts and subtraction factsfor that aggregate sum.

REG. 8 shows boards 2%, 2d, 21, 22 like those in FiGURES 1'1, 1, 2, and3 for teaching aggregate sums 1, 2, 3, and 5 respectively. Associatedwith each board is a container or envelope 61 in which to keep the tilesneeded :to make the groupings of aggregates to teach the addition fact-sof the aggregate sum being taught by that set. FIG. 8 shows the bookopen at the set for teaching the aggregate sum 5 and the tiles 44 to 48inclusive are shown partly removed from the envelope 61.

It may be noted, that when learning the addition or subtraction factswith regard to the number or aggregate sum 1, only one tile 44 havingone uni-t is needed and'the workspace 39 also is equal to only one unit.See FIG. 11. However, in this specification a tile having only one unitis considered an aggregate and is treated in the same way as aggregateswhich are composed of two or more of the basic units. The sets forteaching odd numbers require the same number of-tiles as the numberrepresenting the aggregate sum being taught. Thus aggregate sum 3requires 3 tiles 4 4, 45 and 46. Aggregate sum 5 requires 5 tiles 44, 45, as, 47 and 43.

Sets teaching even numbers require one additional tile in view of thefact that two tiles are needed having half the units of the workspacei.e. of the aggregate sum.

Glbviously in the apparatus such as that shown in FIG. 8, there could beprovided a workspace for every consecutive number or aggregate sum.

The set for number or aggregate sum 4, which-would go in between thesets shown in FIGS. 2 and 3, would comprise aboard having a workspaceshaped like the tile 67. The set would include 5 tiles, namely, aone-unit tile 4 a three-unit tile 46 and two'two-unit tiles 45.

Similarly between the set for teaching the aggregate sum 5 shown in FIG.3 and the set for teaching the aggregate sum 8 shown in FIG. 4 therewould he sets for the aggregate sums 6 and 7 respectively.

In the drawings FIGS. 14 illustrate only selected aggregate sums but aset of the apparatus to teach all the elementary addition facts wouldcover aggregate sums from 1 to 18, or at least 1 to 9.

FIG. 9 is a plan View of an assortment of 18 different tiles 44 to 51and oz to 7 1. inclusive, namely tiles having 1 to 18 units respectivelyand each corresponding in size, shape and arrangement to the arrangementof the units which a workspace for teaching the aggregate sums l to 18respectively would have. A complete set of tiles for learning all theelementary addition facts about the number 13 would include two tileslike tile 62 representing the number 9.

FIG. 10 shows a chart which may be used as a SOIL of final check or aidfor the pupil or the teacher to be sure that the pupil has acquired'afixed knowledge of'the maximum number of relationships of the abstractnumber symbols which it is so necessary to substitute for the count ingsystem in order to operate in the mathematics field with facility.

The chart shows a summary of addition facts to be taught by means ofapparatus similar to that shown in FIGURES 1 to 4, 9 and 11 inclusive.Numerals 74along the top line of FIGURE 10, represent the aggregate sumsto be taught, is. 1 to 18 inclusive. 7

In a column immediately below each aggregate sum are listed the additionfacts to be taught about that aggregate sum.

The numeral in the diagonal line 75 at the bottom of the column undereach aggregate sum is a figure representing the number of tiles neededin the teaching of the simple addition or subtraction facts about thataggregate sum. For example, for aggregate sum 1, 1 tile is needed. Toteach each of the aggregate sums 2 and 3 a total of 3 tiles will beneeded and so forth.

It has been found that it frequently is helpful to the pupil if thenumber symbol of the aggregate sum being learned is put in the workspaceof the final one of the area units formed by these partition lines.Thus, for example, in FIG. 1 the numeral 2 is found in the right handunit of the workspace while in FIGS. 2 and 3 the numerals 3 and 5respectively, are shown in the single lower most basic area unit of theworkspace in those two figures. These numerals are designated by thereference characters 45, 41 and 42. They may be located elsewhere. Ifdesired, other symbols such for instance, as the letters A, B and C canbe substituted for the number symbols used in the drawings of FIGS. 1, 2and 3. This may be useful if the apparatus herein described withrelation to addition and subtraction is to be used in teaching morecomplex mathematical facts.

Elementary addition and subtraction facts must be learned differentlyfrom multiplication which later process or operation deals solely with avarying number of units of like size. In learning simple addition andsubtraction facts, the problem is conceived of as consisting in dealingwith aggregations of different numbers of units, and consequently withtiles of different size. One of the disadvantages of trying to teachaddition and subtraction facts by tallying or counting on ones fingersis that .it is apt to concentrate on using only like units. Thisinvention avoids that disadvantage. While the basic unit is always thesame, tiles or aggregates have varying quantities of basic units. Thearea of each unit is the same but the tiles have diiferent area sizes.

By the use of applicants educational apparatus the pupil learns therelation of these different amounts of units to each other and finallythe relations to the number sym bols in which mathematical operationsare carried out.

One of the characteristics of applicants apparatus is what applicantterms groupings. Thus a grouping of aggregates equals the sum, oraggregate sum, being learned and when the equality is stated one has astatement of addition or subtraction fact. There are of course amultiplicity of addition and subtraction facts for each aggregate sumbeing learned, whether that aggregate sum happens to be part of anaddition fact or a subtraction fact. For example, for the sum of 1 thereis a group of addition facts 201 and a group of subtraction facts 202 inFIG. 11. The group of addition facts represented by the referencecharacter 31 for the aggregate sum of 2 and the group of subtractionfacts represented by the reference character 35 in FIG. 1 each includethree facts. The addition facts for the aggregate sum of 3 in FIG. 2contain four statements marked by the number 32 and the subtractionfacts for that same aggregate sum in FIG. 2 are marked with thereference character 36. Each group of facts in this figure contains fourstatements.

FIG. 3 which is the preferred form of equipment for teaching theaggregate sum of 5 contains addition facts 33 of which there are six andsubtraction facts 37 of which there are also six. Each statement of factconsists of a grouping 72, an equal sign, and the answer 73. Theequality sign, while useful, may not always be essential as long as itis explained to the pupil in some way or other that there is equalitybetween the grouping and the answer.

As thus far explained, the apparatus involves the workspace, the tilesof different size each called an aggregate, the tiles being of such sizeand shape that each tile, when placed with not more than one otherparticular tile of the set for the aggregate sum being learned, willfill the workspace. The tiles necessary to fill the workspace constitute a grouping of the aggregates thereby suggesting the equality ofthe grouping with the aggregate sum. The Workspace is filled by not morethan two tiles. When the statement of fact being learned involves azero, only one tile is required to fill the space, i.e. a tile havingthe same number of units and the same size and shape as thecorresponding workspace. Thus for example, in FIG. 1 the two-unit tile45 fills the entire workspace and is the aggregate 2 in the grouping 2+0and in the grouping 0+2, as Well as the aggregate sum 2.

Further to assist the pupil in learning the addition facts from thetiles and from them the number symbols, it is desirable not only to havethe basic unit area of a simple shape such as a square but also toarrange the tiles involving more than one unit area in a regular manner.Thus in the example shown in the drawings the arrangement of the tileswhich have more than one unit of area has been made such that the pupilwill also learn the differences between odd and even numbers. Thisarrangement of the area units shown in the examples in the drawings alsoconveys to the pupil a sense that all even numbers are similar in somerespects and that all odd numbers also have similar aspects.Furthermore, this arrangement serves to facilitate the recognition ofthe tiles needed to fill the workspace. The arrangement employed in thedrawings for the location of unit areas in the tiles having more thanone unit of area consists essentially in arranging two unit areas sideby side, and if there are more than two area units, placing them belowthe first pair in not more than a pair per level. If the number of areaunits is odd, the odd unit is placed below the lowermost pair and alwayson the same side. In the case of the example shown in the drawings, theodd unit, if there is one, is placed under the left unit of the bottompair. Thus, for example, in FIG. 2 where the aggregate sum 3 is beinglearned, the workspace contains three basic area units, two arrangedside by side and the third below the left hand unit. The tile 45 in thisfigure corresponds to the workspace arrangement, although in use a tilemay fall in any position and the child may have to turn it to the properposition to fill the workspace.

FIG. 3 shows a set for teaching the aggregate sum 5 and thecorrespondence between the workspace and the tiles is carried forward.The set shown in this figure includes a workspace of five units, withtiles of one unit,

of two units, of three units like those shown in previous figures, andalso four units and tile 48 with five basic units. In each case the oddarea unit or representation is under the left one of the lowermost pairin each tile. The tiles 45 and 47 having an even number of area unitsare even on the bottom and this differentiation between odd and even hasbeen found to be a facile method of recognition for the pupil. Thus byusing a small square for each unit with the total area equal to anygiven aggregate always arranged in the same shape and that shape alwayspresented by the workspace in the same position, concrete physicalrepresentation fosters comprehension. It also fosters the idea thatcertain aggregates are odd numbers and certain aggregates are evennumbers, as any representation which has an uneven bottom is an oddnumber.

To date the element of the invention known as the workspace and theelement defined as the tiles have been discussed. The number symbolelement which can be included in the statements of addition andsubtraction facts will now be described in more detail. The workspaceand the tiles have prepared the pupils mind for the grouping and thesestatements of fact and generally a teacher is able to establish therelation between the concrete representations, the workspace and thetiles on the one hand and the corresponding number symbols in thegroupings and the resulting statements of addition and subtraction factson the other. By arranging the statements of addition facts in columnarform, a pupil can see that there are series of addition facts which leadto the same aggregate sum. The similarity of this abstract material tothe concrete workspace and pairs of tiles forms a connecting linkbetween the readily understood concrete groupings and abstractly statedaddition facts. The relation between the tiles and the Workspace and thedifferent addition facts will enable the pupil to sense the significanceof the number symbols with relation to the addition facts. It has beenfound that the subtraction facts are learned through the workspace andthe tiles and then through the groupings and statements of subtractionfacts similarly to the addition situation are sensed and related withfacility.

A manner of using the preferred form of the novel apparatus will now bedescribed.

In the present invention the pupils mind is directed to learning theelementary addition facts or subtraction facts, or both, with regard toone aggregate sum at a time. Nothing concerned with any other sum shouldbe allowed to divert the pupils attention from the particular aggregatesum being learned and the addition and subtraction facts relatedthereto. Actually the pupil is generally taught to learn only oneaddition or subtractionfact even of that particular sum before he learnsthe next fact although generally there is no harm, and in many casessome ad vantage, in having all the addition facts connected with thataggregate sum visible to the pupil at the same time.

The normal'order for teaching different aggregate sums is to start withone and progress upwardly.

Usually a child old enough to be in school will have a good idea of thequantity of one and will also understand the meaning of none, nothing orno, all of which are ways of expressing the idea of zero.

With beginners the teacher can teach the sum one and the additionand'subtractionfacts concerning one, namely, one and'nothing are one andnothing and one are one by concrete examples and by acting. Forinstance, with any object the teacher can show the object and say, Thisis one and can then demonstrate that one plus nothing equals one. Thiswill be plain to the student. Thence mal child of school age has a clearidea of itself as one person. He will readily see that he is one child;one child and no other is still one child.

If the child of school age lacks the idea of himself as oneperson, apartfrom his'school groupor from his family group, there is evidence that heis either emotionally or mentally retarded. Under these circumstances,making the child aware of himself, apart from his group, is important toavoid later confusion in all subjects, not only in mathematics. It hasbeen found, especially in teaching retarded children, that to avoidlater confusion in mathematics it is important to teach the concept ofone and the basic addition and subtraction facts in regard to one beforegoing on to any other aggregate sum.

As soon as the addition facts of oneare understood by means of concreteobjects, the pupil can proceed to the abstract representation. Hewill'have learned the name symbols of the addition facts from oralpresentation by the teacher. If he has not already learned the abstractnurneral sysmbols 1 and 0, he can be taught to read and write them atthis stage.

When the pupil has learned the aggregate sum one, he is ready to go onto the next aggregate sum two. Within the beginning student each newaggregate sum should be demonstrated by means of well known objects.From these he can go on to accept the concrete tiles 44 as units. Theword unit need not be mentioned, nor need most of the words which areused in this specification to describe the novel apparatus.

Working with the apparatus shown in FIG. 1 of the drawing, the teacherwill first take up the tile 44 and say, This is one then tile 45 andsay, This is two. Then pointing to the concrete workspace 26, indicatingboth units, will say, This is two and finally indicating the numeral 2'(reference numeral 46), will say, This is 2. The teacher will then takeup the 2-unit tile 45' and say, Two and nothing are two and place thetile 45 in the workspace 26, the act being a concrete example of thegrouping of the abstract terms 2+0=2. The teacher will then remove tile45 and put it aside and will take up the two one-unit tiles 44 and willsay and demonstrate to the pupil, One and one are two, placing the tilestogether side by side in the two-unit workspace 26. After one or morerepetitions the teacher will encourage the pupil to go through the sameprocedure. Demonstrating the third addition fact, the teacher will say,Nothing and two are two, (taking up tile 45) and will then place thetwo-unit tile 45 in the two-unit Workspace 26.

When the child isthoroughly familiar with the concrete aspect of theaddition facts in connection with the oral presentation of them, theteacher will direct his attention to the successive addition factstatements 31 which, in FIG. 1, appear on the left .side of theworkspace 26, thus teachinghirn to read these concrete statements'aloudand willthen teach him to write the corresponding statements in numbersymbols.

Similarly the child will learn the subtraction facts. When the two-unittile is in workspace 26, the teacher will point to the tiles and say,This is two and then will say, for example, I take nothing away, twoless nothing is two.

Then the teacher will remove the two-unit tile 45 without comment andwill put two one-unit tiles 44 into the workspace 26, also withoutcomment. Then the teacher will'remove one of the one-unit tiles, leavingthe other in the workspace 26 and will say, Two take away one leavesone.

Then replacing the tiles 44 in the workspace, the teacher will removeboth tiles 44 and will say, Two take away twoleaves nothing. Variationsof that subject can be repeated by putting the two-unit tile in theworkspace, removing it, and then saying, Two take away two leavesnothing. Nothing is in this space. The pupil performs the operationhimself too.

When the pupil has arrived at the point where he knows abstractly allthe addition and subtraction facts for any given aggregate sum, theboard containing those state ments of the addition and subtraction-factsfor that aggregate sum is returnedto the teacher, and the pupil will beintroduced to a set of a new board and tiles for a next aggregate sumwhen he has learned the precedingone.

When the aggregate sum 4 has been learned, the pupil will go on theapparatus shown'in- FIG. 3 for teaching the aggregate sum 5.

In teaching the. aggregate sum 5, the teacher will take the tile 48 andshow it to the child and say, This is five. Pointing to the workspace'28, the teacher will say, This is five. Then taking tile 4-8, theteacher will say, Five and nothing are five, and either place the tile48 in the workspace 28 or let the child perform the recital and the act.The teacher can then also indicate the first mathematical statement inthe list 33 of FIG. 3 and pointing to the abstract numerical symbol sayagain, Five and nothing are'five.

With each mathematical statement it is desirable that the child work onthat statement alone until he has mastered it, both in the'concrete andthe abstract.

For the sake of the future understanding of mathematics, it is importantthat the child concentrate on one fact only until it is'learnedthoroughly before he goes on to the next fact. With bright orprecocious'children this may be almost instantaneous. Some of the moreadvanced students can go on immediately and by themselves correlate theconcrete and abstract facts, namely, fitting the four-unit tile 17 andthe one-unit tile 44' into the workspace 23;

With slower children, the child may have to spend some time on thegrouping in order to fully learn first the concrete fact and then toassociate it with the abstract representation of the fact.

With the slower child, the teacher will demonstrate each one of theaddition facts of the aggregate sum 5 by means of the several groupingsof two tiles in the work space 28.

By thus doing the addition facts in close relation to the concreterepresentations of the tiles, in addition to'this 9 manipulation and theoral discussion with the teacher, the pupil will be able to grasp therelation of the abstract number symbols to the concrete symbols.

The number symbols used in the apparatus are preferably numerals, butthe corresponding words may also be used as number symbols.

If further assistance is needed to substitute orderly perception of themeaning of the groupings and the addition and subtraction facts for themeaningless counting learned by rote, it is possible to use theconstructions shown in FIGS. 5, 6 and 7 of the drawings. Here, inaddition to the concrete facts or elements of the workspace and tiles,and the abstract number symbols, an intermediate element for learning isinterposed, which may be called a semi-concrete element.

In FIGS. 5 and 7 a sheet 52 is shown as a semhconcrete element. Thesheet is shown in FIG. 7 as it would normally be given to the child,with three blank representations 55 of the concrete workspace 28 shownin FIG. 3, none of which has been filled in. The child is asked to colorthe representations to show the varying combinations of tiles which fitinto and fill the workspace. FIG. 5 shows a similar sheet 52 on whichthe child has filled the representations with color to show thearrangements of tiles which represent the different addition andsubtraction facts. One representation is lined to show that the childhas filled it in with red to represent a five-unit tile 48 or abstractlyto represent the statement 5+0=5 or +5 :5. In the same figure the nextspace has been filled in with red to indicate a four-unit tile 47 andwith green to indicate a one-unit tile 44 and to represent the abstractstatements 4+1=5 or l+4=5. The child has indicated in red at the thirdunit a two-unit tile 45 and in green a three-unit tile 46 to representthe abstract statements 3+2=5 and 2+3=5. It may be preferred not to haveany number symbols in the workspaces.

In the process of teaching or learning the concrete elements of theapparatus and their use, the pupil will be led to color, i.e. fill inthese squares in each blank representation of the workspace tocorrespond with the concrete tiles needed to fill the concreteworkspace. The number of representations of the workspace provided isequal to the number of addition facts for the aggregate sum beinglearned but preferably eliminating any duplication of facts such as merereversal of the order of the aggregates in the groupings of thestatements. The pupil will learn to color or mark in the combinations oftiles corresponding to the aggregates which will give the correctaggregate sum or answer.

FIG. 6 is an embodiment of another modified form of a separate sheet 24for use in connection with the quantity or sum 5. It differs from thesurface 22 shown in FIG. 3 in that the representations of subtractionfacts are omitted, and the representations 56 of the addition facts areshown in words.

If the element being learned is the written or printed words whichrepresent the numbers, those words having been heard orally indiscussion with the teacher can be mentally associated in groupings togive equality to the aggregate sum being learned.

In FIG. 7 the addition and subtraction facts are not directly on thesurface of the board 25. Cooperating removable sheets 58 and 59 areprovided embodying the respective addition and subtraction facts. Therealso is shown a removable sheet 52 embodying semi-concreterepresentations 55 of aggregate sum similar to that shown in FIG. 5,detachably associated on the surface 25. These elements of the workspaceand the numerical statements of the addition and subtraction facts areon portions of the board 25 which are detachable from each other. Theyare shown fastened together by simple tongues 53 and grooves 54. The useof detachable means makes it possible to rearrange the order of theelements or to omit one or more as necessary. Any sheet or part of the1% board removed is shown punched for transfer to the pupils ring book60, shown in FIG. 8.

From what has already been said, those skilled in the art willappreciate that as the facts of each new aggregate, quantity or sum istaught, a few new addition and subtraction facts are added to thosewhich have already been taught until at last when 18 is the aggregatesum, the pupil will have learned all the addition facts about all thepreceding aggregate sums l to 17 inclusive. The initial concreterecognition of the quantity 18 may or may not be new at this time. It ispossible that before this the pupil will have learned that there is aquantity 18 by means of counting objects.

Many of the addition facts for this aggregate sum will be learned almostautomatically by analogy with the addition facts for the preceding sums.Probably the learning of 18+0=18, 17+1=18, 16+2=18, l5-1-3=l8,14+4=l8,13+5=18,12+6=18,11+7=18,10+8=18 will be relatively simple. Theonly really new thing in teaching the aggregate sum 18 is 9+9=18. Thisis the critical addition fact in connection with the aggregate sum 18and it is the last of the simple addition facts which the child needs tolearn before going on into multiple addition or simple multiplication.

It will have been noted that in the preferred embodiment of theapparatus the basic unit is always shown in the same size'and shape ineach aggregate sum from 1 to 18. Also any aggregate sum for eachquantity 1 to 18 is always shown in the same size and shape wherever itis met, and the relation of the units is the same each time it appears.

It has been found useful in elementary presentation of the simpleaddition and subtraction facts to begin with the concrete and lead tothe abstract as set forth above.

It will have been seen that from one point of view the object of theinvention is to teach arithmetic on an abstract level with specialclarification of addition and subtraction. This is important becausetallying or countingusing stones, sticks, fingers or an abacus-for toolong delays and blocks the majority of pupils from progressingsatisfactorily in mathematics beyond its primary forms. Often theteacher heretofore has attempted to jump from counting to teachingabstract mathematical statements by rote rather than by helping thepupil to understand" the relation between the concrete objects, theabstract number symbols and the abstract expressions of addition andsubtraction facts.

The child should firmly grasp the relation between the concrete andabstract, and between the cardinal and ordinal meanings of numbersymbols, e.g. 5 as an aggregate rather than the fifth unit in counting,before going on to multiple additions, and to multiplication as aprocess dealing with identical aggregates.

In the novel apparatus the workspace and the separate tiles teach thatthe unit is a recognizable part of each aggregate and of each aggregatesum. The workspace of the aggregate sum one and the tile 44 show thatone is a. unit. The pupil is thus led to learn that aggregates of unitsare definite quantities. He also learns that the workspace represents acombination of quantities, which lead him into the subject of groupings.The concept that the one or two tiles fill the entire workspace relatesthe units and the aggregates to the aggregate sum, all this giving moremeaning to the addition facts. The perception and meaning of thesubtraction facts are taught along with the addition facts as theirinverse. If the abstract number relations are learned in this manner,the idea of counting as the only meaning attached to the numbers can becounteracted if not substituted for. Including the number symbols inwords or numerals in writing in the apparatus generally is sufficient towork in with the oral use of number symbols in words as given by theteacher and used by the pupil to enable the pupil to proceed from theconcrete workspace and tiles to the abstract number symbols.

amines This invention is particularly useful with those who havecomparatively little knowledge of arithmetic such, for instance, asmentflly retarded children, although its usefulness for any one will beclear from the foregoing description.

While it is preferred to arrange the units as shown in the workspacesand tiles illustrated in the figures of this application. units of othershape and workspaces of other dimensions can be used.

Other embodiments of the invention will occur to those skilled in theant.

What is claimed is:

1. Educational apparatus for the teaching of elementary addition orsubtraction facts about a consecutive series of cardinal numbers,comprising (a) a plurality of workspaces each of a different areacorresponding to a ditferent cardinal number in said series, but allbased on the same area unit,

([2) in combination with tiles whose areas are based on the same unit,

() there being a set of tiles for each workspace, in-

cluding (d) a tile of the same size and shape as said workspace andother tiles of lesser size than the workspace,

(e) not more than two of said lesser tiles having the same size andshape,

(7) and each lesser tile being of a size and shape such that when put ina grouping with one other lesser tile fills the workspace,

(g) in which the number of lesser tiles for each set equals not morethan the cardinal number about which facts are being taught and notfewer than one less than said number; whereby each elementary additionfact for each of said cardinal numbers can be taught by putting not morethan two tiles in the workspace corresponding to said number.

2. Educational apparatus according to'claim 1 in which each workspace ison a separate surface.

3. Educational apparatus according to claim 1 in which the consecutiveseries of cardinal numbers represented is within the numbers 1 to 18.

4-. Educational apparatus according to claim 1 in which each workspaceand tile is visibly marked into said area units.

5. Educational apparatus for the teaching of elementary addition orsubtraction facts about a consecutive series of cardinal numbers,comprising (a) a plurality of workspaces each of a different areacorresponding to a diiferent cardinal number in said series, but allbased on the same area unit,

(b) in combination with tiles whose areas are based on the same unit,comprising (c) at least two tiles the same size and shape as eachworkspace corresponding to a cardinal number in the lower half of theseries and (d) at least one tile the same size and shape as each of theother workspacesin the series, said tiles providing, for any cardinalnumber in said series, a set of tiles for the workspace for said numberrepresenting each elementary addition fact for said number, said setconsisting ofa tile of the same size and shape as the workspace andother tiles of lesser size than said workspace, such that each lessertile when put in a grouping with a single additional lesser tile fillsthe workspace, not more than two of the said lesser tiles being of thesame size and shape.

6. Educational apparatus according to claim 5' in which each workspaceis on a separate surface.

7. Educational apparatus for the teaching of elementary addition orsubtraction facts about a cardinal number, said apparatus comprising (a)a plurality of workspaces each of a different area corresponding to adifferent cardinal number in a consecutive series, but all based on thesame area unit,

(b)- at least one of said workspaces having its area visibly marked intoarea units totaling the cardinal number about which facts are beingtaught, in combination with (c) tiles whose areas are based' on the sameunit, and

are visibly marked into said area units,

one tile being of the same size and shape as said workspace and theother tiles of lesser size than said workspace,

not more than two of said lesser tiles having the same size and shape,

and each lesser tile being of a size and shape such that when put in agrouping with one other lesser tile fills said workspace, and

in which the total number of lesser tiles for said workspace equals notmore than the cardinal number about which facts are being taught and notfewer than one less than said number,

whereby each elementary addition fact for said cardinal number can betaught by putting not more than two tiles in said workspace.

8. Educational apparatus according to claim 7 (d) in which there areassociated with said workspace statements of the elementary addition orsubtraction facts for said cardinal number, and

(e) a plurality of semi-concrete representations the same size and shapeas the work space and visibly marked the same as the workspace andadapted to be further marked by the learner to show tile groupings orelementary mathematical facts which relate to the cardinal number,

whereby a learner can progress from a concrete to an abstractunderstanding of said number.

References Cited in the file of this patent UNITED STATES PATENTS356,167 Shannon Jan. 18, 1887 1,528,061 Joyce Mar. 3, 1925 1,826,034Williamson Oct. 6, 1931 1,836,870 Quer Dec. 15, 1931 2,866,278 SnarrDec. 30, 1958

1. EDUCATIONAL APPARATUS FOR THE TEACHING OF ELEMENTARY ADDITION ORSUBTRACTION FACTS ABOUT A CONSECUTIVE SERIES OF CARDINAL NUMBERS,COMPRISING (A) A PLURALITY OF WORKSPACES EACH OF A DIFFERENT AREACORRESPONDING TO A DIFFERENT CARDINAL NUMBER IN SAID SERIES, BUT ALLBASED ON THE SAME AREA UNIT, (B) IN COMBINATION WITH TILES WHOSE AREASARE BASED ON THE SAME UNIT, (C) THERE BEING A SET OF TILES FOR EACHWORKSPACE, INCLUDING (D) A TILE OF THE SAME SIZE AND SHAPE AS SAIDWORKSPACE AND OTHER TILES OF LESSER SIZE THAN THE WORKSPACE, (E) NOTMORE THAN TWO OF SAID LESSER TILES HAVING THE SAME SIZE AND SHAPE, (F)AND EACH LESSER TILE BEING OF A SIZE AND SHAPE SUCH THAT WHEN PUT IN AGROUPING WITH ONE OTHER LESSER TILE FILLS THE WORKSPACE, (G) IN WHICHTHE NUMBER OF LESSER TILES FOR EACH SET EQUALS NOT MORE THAN THECARDINAL NUMBER ABOUT WHICH FACTS ARE BEING TAUGHT AND NOT FEWER THANONE LESS THAN SAID NUMBER; WHEREBY EACH ELEMENTARY ADDITION FACT FOREACH OF SAID CARDINAL NUMBERS CAN BE TAUGHT BY PUTTING NOT MORE THAN TWOTILES IN THE WORKSPACE CORRESPONDING TO SAID NUMBER.